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Theorem aev 1890
Description: A "distinctor elimination" lemma with no restrictions on variables in the consequent. (Contributed by NM, 8-Nov-2006.) Remove dependency on ax-11 1791. (Revised by Wolf Lammen, 7-Sep-2018.) Remove dependency on ax-13 1968, inspired by an idea of BJ. (Revised by Wolf Lammen, 30-Nov-2019.)
Assertion
Ref Expression
aev  |-  ( A. x  x  =  y  ->  A. z  w  =  v )
Distinct variable group:    x, y

Proof of Theorem aev
Dummy variable  u is distinct from all other variables.
StepHypRef Expression
1 aevlem1 1886 . . 3  |-  ( A. x  x  =  y  ->  A. u  u  =  w )
2 ax6ev 1721 . . . 4  |-  E. u  u  =  v
3 ax-7 1739 . . . . 5  |-  ( u  =  w  ->  (
u  =  v  ->  w  =  v )
)
43aleximi 1632 . . . 4  |-  ( A. u  u  =  w  ->  ( E. u  u  =  v  ->  E. u  w  =  v )
)
52, 4mpi 17 . . 3  |-  ( A. u  u  =  w  ->  E. u  w  =  v )
6 ax5e 1682 . . 3  |-  ( E. u  w  =  v  ->  w  =  v )
71, 5, 63syl 20 . 2  |-  ( A. x  x  =  y  ->  w  =  v )
8 axc16g 1887 . 2  |-  ( A. x  x  =  y  ->  ( w  =  v  ->  A. z  w  =  v ) )
97, 8mpd 15 1  |-  ( A. x  x  =  y  ->  A. z  w  =  v )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1377   E.wex 1596
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-12 1803
This theorem depends on definitions:  df-bi 185  df-ex 1597
This theorem is referenced by:  ax16nf  1891  axc16gALT  2079  2ax6e  2180  wl-naev  29575  wl-ax11-lem2  29619
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